Assume for a moment that a set \(A:=F\)  is closed, and that a measure \(\mu :=\sigma \)  is of finite energy, i.e. \(\sigma \in \mathcal {E}^+_\alpha \) . Based on the facts that the \(\alpha \) -Riesz kernel is perfect and satisfies the domination principle, one can prove by generalizing the classical Gauss variational method (see [1]}, [2]}, cf. also [3]}) that there exists \(\sigma ^F\in \mathcal {E}^+_F\)  uniquely determined within \(\mathcal {E}^+_F\)  by the equality \(\kappa _\alpha \sigma ^F=\kappa _\alpha \sigma \)  n.e. on \(F\) .
This \(\sigma ^F\)  is said to be the \(\alpha \) -Riesz balayage of \(\sigma \in \mathcal {E}^+_\alpha \)  onto (closed) \(F\) , and it can alternatively be characterized as the orthogonal projection of \(\sigma \)  in the pre-Hilbert space \(\mathcal {E}_\alpha \)  onto the convex, strongly complete cone \(\mathcal {E}^+_F\)  (see Sect. );
that is,Concerning the orthogonal projection in a pre-Hilbert space, see e.g. [4]}.
\(\Vert \sigma -\sigma ^F\Vert =\min _{\nu \in \mathcal {E}^+_F}\,\Vert \sigma -\nu \Vert .\) 
